The Physics of a Falling Slinky

Shimon Kolkowitz
October 31, 2007

Introduction

Using a simple household toy it is possible to
observe a strange and intriguing physics phenomenon. If a slinky is hung
by one end such that its own weight extends it, and that slinky is then
released, the lower end of the slinky will not fall or rise, but will
remain briefly suspended in air as though levitating. [1] To achieve the
full visual effect, one may wish to attach a brightly colored object to
the lower end of the slinky. [2] This visually impressive display can be
modeled by treating the slinky as a tightly wound spring. [3] By
applying Hooke's law in conjunction with Newton's second law, it is
possible to solve for the static equilibrium of a hanging Slinky, and
then use this result to solve for the motion of the Slinky as it falls.
In particular, I will derive an expression for the "levitating time"
over which the bottom of the Slinky remains motionless, and compare it
to an experimentally measured value for a metal Slinky.

The Hanging Slinky

A hanging Slinky may be treated as a hanging spring,
but with a few qualifications. Real springs, including Slinkies, have
coils of finite thicknesses, so that there is a minimum compressed
length at which the coils are touching. Indeed, as opposed to an ideal
spring, a Slinky is usually pretensioned, meaning that in the absence of
an external force the coils are touching, and a finite force f = k
(l1 - l0 ) is required to cause the coils to
separate, where k is the spring constant, l1
is the minimum compressed length of the Slinky, and lo
is the unachievable zero force length. [4]

To solve for the motion of a falling Slinky, we use
a dimensionless variable d to denote points on the Slinky, where d =
0 denotes the top end of the slinky, and d =1 denotes the
bottom. The difference in d between any two points on the slinky
is equal to the fraction of the Slinky's total mass between those
points. The position and motion of the Slinky is described by giving the
location y(d,t) of all points d of the slinky as functions
of time t. [3] For the hanging slinky, we find
y0(d), which is static and not dependent on t.
Above a certain point d1 the Slinky behaves like a
loosely wound spring according to the equation

(1)

Which is a direct result of Hooke's
law. [4] Because the Slinky is pretensioned, below d1
the coils are in contact, because the weight of the Slinky below that
point is not enough to overcome the finite force f. [4]

By setting d = d1 in equation 1, and
adding l1 (1 - d1) (the length of the section
below d1), we obtain the total suspended length of the
slinky:[4]

(2)

Using equation 2, it is easy to experimentally
determine the spring constant k of a Slinky by measuring its
collapsed length l1, its suspended length
yo(bottom), its mass m, and
d1. I measured these values for a metal Slinky
purchased at a local toy store in order to determine its spring constant
k. My measurements are listed in Table 1. I obtained k =
0.80 &plusmn 0.04 kg/s2 for the Slinky.

The Falling Slinky

Having derived an expression for the static
equilibrium of a hanging slinky, let us now suppose that at time t =
0 the top of the slinky is released. One coil after another begins
to collapse down, with the coils below the leading edge of this collapse
remaining motionless. The bottom of the Slinky appears to levitate in
the air, remaining in the same spot until the leading edge of the
collapsing portion of the slinky reaches d1, at which
point the whole slinky falls to earth in a collapsed coil. I
experimentally measured the "levitating time" of the metal Slinky, and
found it to be t = 0.4 &plusmn 0.1 seconds.

Table 1: Measurements for toy
Slinky.

Mass m

Suspended Length y0

Collapsed Length l1

Fraction Coiled d1

0.218 ± 0.001 kg

1.365 ± 0.003 m

0.057 ± 0.003 m

0.99 ± 0.005

As the Slinky falls, there are three portions
of the slinky. The bottom portion, below d1, remains
motionless and coiled together. The middle section, above
d1 and below the leading edge of the collapse is yet to
be affected by the fall and remains governed by equation 1. The locations
of the points in the top portion, above the leading edge of the collapse,
are given by:

(3)

Where d(t) is the location of the leading
edge of the collapsed top section. [3] We get this result by realizing
that the derivative of y(d,t) with respect to d for the
top section must be l1 , because this section is
collapsed, and from the boundary condition that y(d,t) for the
top section must be equal to y(d,t)for the middle section at the
point d=d(t). [3] If we take the derivative of equation 3 with
respect to time, we get the velocity of the top collapsed section:

(4)

The mass of the top portion of the slinky is given
by md(t), and as this portion of the Slinky is the only portion
in motion, the total momentum of the slinky is given by its mass times
equation 4. Using Newton's second law, we can equate this momentum to
the total impulse mgt that has acted on the system to get
[3]:

(5)

By integrating equation 5 we get an equation for
d(t):

(6)

Now, by plugging d(t) = d1 into
equation 6, we find the time it takes until the Slinky collapses fully,
and the bottom begins to move [3]:

(7)

Plugging the experimentally measured values for
m and d1, and the value k calculated
using equation 2, this gives a "levitating time" for the metal Slinky of
t = 0.29 &plusmn 0.05 seconds. This corresponds relatively well
to the experimental value of t = 0.4 &plusmn 0.1 seconds.

Conclusion

When a Slinky is dropped, the bottom of the Slinky
remains motionless as the top collapses towards it, making it appear to
the observer as though the Slinky is levitating. By considering the
Slinky as a tightly wound, pretensioned spring, the static equilibrium
of a hanging Slinky was solved for using Hooke's law (Equation 1). This
result was used to measure the spring constant of an actual metal
Slinky. The motion of the Slinky after it is released at time t=0
was then solved for to derive an expression for the time over which the
bottom of the Slinky remains motionless and the Slinky appears to
levitate (Equation 7). This expression gave a value of t = 0.29
&plusmn 0.05 seconds for the Slinky used in the experiments, which
matches up very well with the experimentally measured value of t =
0.4 &plusmn 0.1 seconds.